The inversion of fractional integrals on a sphere

The purpose of the paper is to invert Riesz potentials and some other fractional integrals on the n-dimensional spherical surface in ℝ ⁿ⁺¹ in the closed form. New descriptions of spaces of the fractional smoothness on a sphere are obtained in terms of spherical hypersingular integrals. It is shown that Riesz potentials of the orders n, n + 2, n + 4, ... on a sphere are Noether operators and their d-characteristic depends on the radius of the sphere.